Question: Simplify and expand the following expression: $ \dfrac{p + 7}{5p - 4}+\dfrac{p + 9}{p - 6} $
Answer: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(5p - 4)(p - 6)$ Multiply the first term by $\dfrac{p - 6}{p - 6}$ $ \begin{align*} \dfrac{p + 7}{5p - 4} \times \dfrac{p - 6}{p - 6} & = \dfrac{(p + 7)(p - 6)}{(5p - 4)(p - 6)} \\ & = \dfrac{p^2 + p - 42}{(5p - 4)(p - 6)}\end{align*} $ Multiply the second term by $\dfrac{5p - 4}{5p - 4}$ $ \begin{align*} \dfrac{p + 9}{p - 6} \times \dfrac{5p - 4}{5p - 4} & = \dfrac{(p + 9)(5p - 4)}{(p - 6)(5p - 4)} \\ & = \dfrac{5p^2 + 41p - 36}{(p - 6)(5p - 4)}\end{align*} $ Now we have: $ = \dfrac{p^2 + p - 42}{(5p - 4)(p - 6)} + \dfrac{5p^2 + 41p - 36}{(p - 6)(5p - 4)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{p^2 + p - 42 + 5p^2 + 41p - 36}{(5p - 4)(p - 6)} $ $ = \dfrac{6p^2 + 42p - 78}{(5p - 4)(p - 6)}$ Expand the denominator: $ = \dfrac{6p^2 + 42p - 78}{5p^2 - 34p + 24}$